🏅️ Half a century ago, an equation called the Black–Scholes formula revolutionized finance, leading to a rapid growth of markets and stimulating quantitatively oriented minds. However, with time, its simplicity became a liability — and yet its legacy persists. In this blog, we shall journey back to the birth of the Black-Scholes model, exploring its origins, its profound implications, and its enduring legacy.


1 Setting the Stage for a Market Revolution in ’70s

In the Journal of Political Economy’s May-June 1973 edition, Fischer Black and Myron Scholes unveiled a pioneering financial model, a keystone in mathematical finance, which subsequently shaped the operations of financial markets. This influential piece followed Robert Merton’s key advances, shared with Scholes in the 1997 Nobel Prize in Economics, two years posthumously for Black. The trio’s groundbreaking formula, known as the Black-Scholes equation, simply yet elegantly defined the price that investors should pay for financial instruments to hedge the volatility inherent in asset investments.

Simultaneously, in 1973, the world witnessed the opening of the Chicago Board Options Exchange, marking the birth of the world’s first platform for trading financial derivatives - options. There are two primary types of options: calls and puts.

  • A call option grants the holder the right, but not the obligation, to purchase an underlying asset at a specified price (the strike price, \(K\)) before a certain date (the expiry date, \(T\)).

    Mathematically, the payoff of a call option, \(C\), at expiry is given by the formula \(C = max(0, S - K)\), where \(S\) is the price of the underlying asset at expiry. This indicates that if the price of the underlying asset exceeds the strike price (\(S > K\)), the option is exercised, yielding a profit of \(S - K\); if not, the option expires worthless, hence the payoff is zero.

  • Conversely, a put option gives the holder the right to sell an underlying asset at a pre-set price before a certain date. The payoff for a put option, \(P\), is represented as \(P = max(0, K - S)\).

It is crucial to remember that these formulae represent payoffs at expiry; pricing the options requires factoring in the time value of money - payoff received today is worth more than the same amount received in the future expiry \(T\) - and the risk characteristics of the underlying asset.

The Chicago Board Options Exchange in 1973

Figure 1.1: The Chicago Board Options Exchange in 1973

The perfectly timed release of the Black-Scholes formula facilitated consensus among market players about fair option pricing, instilling traders with the clarity and confidence to trade in this blossoming market, thus fuelling its growth.

The world was poised for this financial innovation. The American landscape was rife with disillusionment, with the recent cessation of its 20-year-long Vietnam war involvement, ongoing Cold War tensions, and the Watergate scandal climax. The Bretton Woods system, which established the US dollar as the world’s reserve currency linked to gold, was on its deathbed, ultimately leading to an era of floating exchange rates in 1973. Concurrently, an oil crisis shot up oil prices, triggering inflation and destabilising many economies.

Amidst this backdrop of global upheaval and market unpredictability, the Black-Scholes model emerged as a beacon, shifting investment perspectives towards risk management and offering investment control via certain financial securities. The formula, simple yet powerful, calculates an option’s payoff based on the volatile asset price, the predetermined future trading price (the strike), and other standard probability parameters, removing the associated investment risk. The model’s novelty, however, faced initial publication rejections before finally being accepted after significant revisions.

Contrary to past struggles in risk-pricing options, the Black and Scholes model surprisingly proposed an option price independent of the risk tolerance of the buyer and seller, thus generating a universally fair price.

By 1975, nearly all traders valued options through the Black-Scholes model, further facilitated by pocket calculators. The derivatives market, including options, proliferated rapidly in a mere two years, with total capitalisation reaching trillions of dollars. The work of Black, Scholes, and Merton not only catalysed the explosive expansion of derivatives markets, but also sparked further theoretical advancements in the field.


2 The Dawn of Financial Engineering and Quant

The inception of mathematical finance is commonly associated with Louis Bachelier’s 1900 PhD dissertation. However, some argue the pivotal breakthroughs of 1973 marked both the field’s maturity and the genesis of financial engineering. As noted by US economist Andrew Lo at Robert Merton’s 75th birthday celebration, a scientific field can only be deemed mature when it spawns an engineering branch. Post-1973, the pertinence of mathematical finance techniques to options market engineering was incontrovertible.

This shift led to a surge in demand for specialists adept at traversing the technical labyrinth of the field. Initially, these experts were primarily mathematicians, statisticians and physicists, later joined by computer scientists as technology evolved. This birthed a new species of professionals known as quants, or quantitative analysts, leading to the introduction of relevant courses at universities. The endeavours of these quants, coupled with the symbiotic interaction between scientists and financial engineers, significantly contributed to the rapid expansion of mathematical finance literature in subsequent decades.

Fischer Black could be viewed as one of the pioneer quants. Despite his tumultuous academic journey, oscillating between physics, mathematics, and artificial intelligence, he eventually secured a PhD in applied mathematics. Following his collaborative work with Scholes and Merton at MIT, Black joined Goldman Sachs in 1984, where he created an array of financial products and models, staying active in research and banking until his demise in 1995.

Conversely, Myron Scholes demonstrated early economic interest, venturing into stock market investments during his secondary school years. Scholes, guided by Fama and Miller during his MBA, delved into the nascent field of financial economics. In the early 1990s, he worked for the then-extant Salomon Brothers before co-founding the hedge fund, Long-Term Capital Management, with Merton and John Meriwether. Initially, Long-Term Capital Management flourished, but the Asian financial crisis of 1997 and the Russian economy’s 1998 collapse led to an abrupt downfall. With an equity base of $4.8 billion, the fund’s misfortune edged the global financial system towards a disaster, compelling the US Federal Reserve to intervene. A year after Scholes and Merton’s Nobel recognition, their fund became a notorious testament to the investment industry’s inherent risk.


3 Unravelling the Black-Scholes Model

The Black-Scholes Model, a paragon of mathematical elegance, lays a vital groundwork for pricing options, thereby providing a risk-mitigation framework for investments in volatile assets. It builds upon several critical assumptions that are necessary to ensure the accuracy and appropriateness of its application:

  • Markets are perfectly efficient, devoid of transaction costs or taxes, which eliminates the possibility of arbitrage: The absence of arbitrage ensures that no risk-free profits can be generated, meaning all assets are fairly priced, a necessity for the Black-Scholes model to provide a fair option price.

  • Constant and known volatility \(\sigma\) and risk-free rate \(r\): Having these values constant and known helps simplify the model, as they would otherwise introduce further uncertainty, making it much more complex to estimate the option’s fair price.

    Note the volatility used is an assumption of how volatile the underlying asset will be in the future.

  • The returns of the underlying asset follow a log-normal distribution, ensuring that asset prices cannot fall below zero: simplifies the modelling process because a normal distribution has well-known properties that are mathematically tractable.

  • No dividends are paid during the life of the option.

  • Investors can borrow and lend at a constant risk-free interest rate.

  • Perfect hedging is feasible by continuously buying and selling the underlying asset: allow for the creation of a risk-free hedge portfolio, which is a pivotal part of the Black-Scholes model’s theoretical foundation.


The mathematical expression for the Black-Scholes formula for a European call option is given by:

\[ C = S \cdot N(d_1) - K \cdot \exp^{-rT} \cdot N(d_2) \]

Where:

\(C\) represents the price of the call option.

\(S\) signifies the current price of the underlying asset.

\(N(d)\) stands for the cumulative standard normal distribution function. \(d_1\) and \(d_2\) are variables computed as follows:

  • \(d_1 = [\ln( \frac{S}{K}) + (r + \frac{\sigma^2}{2})T] / (\sigma \cdot \sqrt T)\)

    \(N(d_1)\) is the risk-adjusted probability that the option will be exercised in the risk-neutral world, assuming the asset price finishes above the strike price.

  • \(d_2 = d_1 - \sigma \cdot \sqrt T\)

    \(N(d_2)\), on the other hand, is the risk-adjusted probability that the option will be exercised in the real world.

Central to the Black-Scholes model is the notion that the value of an option is the difference between the present value of the expected payoff of owning the option, and the present value of the cost of hedging against potential losses from owning the option. This captures the delicate balancing act between risk and reward inherent in financial decisions:

  • A higher strike price \(K\), a longer time to expiry \(T\), or increased volatility \(\sigma\) elevate both potential reward and risk, thereby augmenting the option’s value.

  • Conversely, a higher risk-free interest rate \(r\) diminishes the option’s allure compared to risk-free investments, thereby decreasing its value.


“Volatility Smile” - John Hull, a celebrated author, highlighted that traders favour the Black-Scholes theory due to its single unobservable parameter linked to volatility. Despite its appeal, limitations became apparent over time. These shortcomings led to the concept of ‘implied volatility smile,’ indicating a reliance of market-implied volatility on option strike prices. The phenomenon of volatility varying with different contract lengths led to extensive research in mathematical finance, focusing on the ‘implied volatility surface.’

The concept of an “implied volatility smile” is a fascinating and nuanced phenomenon in options trading. It was identified following the market crash of 1987 when analysts noted a peculiar pattern - the implied volatility of options was not a constant (as suggested by the Black-Scholes-Merton model), but varied with the strike price, forming a pattern that resembled a smile when plotted. In mathematical terms, as \(\sigma\) represents implied volatility, and \(K\) is the strike price, the implied volatility smile indicates that \(\sigma\) is not a constant, but rather a function of \(K\), hence written as \(\sigma(K)\). This means that implied volatility changes with different levels of strike price, revealing the limitation of the Black-Scholes model, which assumes constant volatility.


4 The Option Market in the Post-Black-Scholes Epoch

Presently, we find ourselves in a post-Black-Scholes era, where the model’s historical significance remains unchallenged, yet some contend that it can potentially distort, rather than clarify, our comprehension of market microstructure. A multitude of research endeavours have been dedicated to enhancing financial models, quantifying associated risks, and deciphering the implications of potential model inaccuracies, acknowledging the inherent imperfections in every model.

The foundational assumptions of Black, Scholes, and Merton were perceived as oversimplified, giving way to more intricate models capable of replicating the ‘volatility smile.’ Modern models tend to facilitate more diversified movements of the underlying asset price than the Black-Scholes equation allows, offering traders a selection of models with stochastic, ‘rough’, or jump-induced volatility.

  • Local Volatility Model - proposed by Bruno Dupire in 1994, it presumes that the volatility of the underlying asset is both time and asset price dependent, leading to the term ‘local volatility’. It allows for the calibration of market-observed option prices and reproduces the entire implied volatility surface, therefore accurately capturing market expectations and sentiments at any given point in time.

    Dupire, B. (1994). Pricing with a Smile. Risk Magazine, 7(1), 18-20.


  • SABR Model - short for Stochastic Alpha, Beta, Rho, was introduced by Hagan et al. in 2002. The model incorporates a stochastic volatility component and is widely used for interest rate options. The model does an excellent job in capturing the market-implied volatilities across different strikes and maturities, an aspect integral to interest rate options. Moreover, it is popular due to its adaptability to negative rates.

    Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing Smile Risk. Wilmott Magazine.


A significant portion of current research is dedicated to training artificial intelligence to price and hedge options autonomously and in a broader range of circumstances than before – Horvath, B., Muguruza, A. and Tomas, M., 2019. Deep learning volatility. arXiv preprint arXiv:1901.09647.

Likewise, substantial efforts are invested in comprehending how ultra-high-frequency trading impacts markets, and how pricing strategies can be fortified to adapt to continuously evolving market conditions – Avellaneda, M. and Stoikov, S., 2008. High-frequency trading in a limit order book. Quantitative Finance, 8(3), pp.217-224.

Despite these shifts in focus, it’s undeniable that the financial markets and research landscape would not have reached their current state without the trailblazing work of Black, Scholes, and Merton.